Simplifying Algebra

What you'll learn

How to recognise like terms and collect them together.
How to simplify algebra using addition, subtraction, multiplication and division.
How to write repeated letters using powers, such as $x^2$ .
How to avoid common sign mistakes in one- and two-mark simplification questions.

What does “simplify” mean?

To simplify an algebra expression means to write it in a shorter, neater form without changing its value.

For example, if you see $x + x + x$ , you do not need to write the letter three times. You can write $3 x$ instead.

Definition

Expression, variable, term, coefficient

A variable is a letter that stands for a number, such as $x$ , $a$ or $n$ .
An expression is a calculation involving numbers, letters or both, such as $4 x + 7$ .
A term is one part of an expression, separated by plus or minus signs. In $5 x - 3 y + 2$ , the terms are $5 x$ , $- 3 y$ and 2.
A coefficient is the number multiplying a variable. In $6 m$ , the coefficient of $m$ is 6.

Key Idea

The main idea

Simplifying algebra is mostly about collecting matching terms and writing multiplication neatly.

Collecting the same letter

When you add the same letter term more than once, count how many of that term you have.

Think of $p$ as one $p$ . So:

$p + p$ becomes $2 p$ .
$p + p + p$ becomes $3 p$ .

Example

Collecting identical letter terms

Simplify $4 x + 5 x - 2 x$ .

Look at the letter in each term. Every term is an $x$ term, so they are like terms.
Collect the coefficients: 4, 5 and -2.
Work out 4 + 5 - 2 = 7.
Put the letter back on the answer:
$4 x + 5 x - 2 x = 7 x$

Common Mistake

Forgetting the invisible 1

A single $x$ means $1 x$ , not zero $x$ . So $x + x + x$ is $3 x$ , not just $x$ .

Terms with letters and numbers

Sometimes an expression has both letter terms and plain number terms.

A plain number term is called a constant because it does not change when the variable changes.

Definition

Constant

A constant is a term with no variable, such as 5, -3 or 12.

You can collect letter terms with the same letter, but constants stay separate from letter terms.

Example

A letter term and a constant

Simplify $k + k + 9$ .

The terms $k$ and $k$ are like terms.
Since $k + k = 2 k$ , replace those two terms with $2 k$ .
The 9 is a constant, so it cannot be combined with $2 k$ .
The simplified expression is:
$k + k + 9 = 2 k + 9$

Tip

Say it out loud

$2 k + 9$ means “2 lots of $k$ , plus 9”. Since you do not know what $k$ is, you cannot add the 9 onto the $2 k$ .

Like terms and unlike terms

Definition

Like terms

Like terms have exactly the same variable part. For example, $3 a$ and $7 a$ are like terms. But $3 a$ and $3 b$ are not like terms.

Here are some examples:

$5 x$ and $- 2 x$ are like terms.
$4 ab$ and $9 ab$ are like terms.
$x$ and $x^2$ are not like terms.
$6 m$ and $6 n$ are not like terms.

This picture shows the idea of sorting terms before collecting them.

Algebra terms sorted into like-term groups before simplifying

Example

Collecting two different types of term

Simplify $7 g + 4 h - 2 g + 3 h$ .

Group the $g$ terms together: $7 g - 2 g$ .
Group the $h$ terms together: $4 h + 3 h$ .
Simplify each group:
$7 g - 2 g = 5 g$
Simplify the other group:
$4 h + 3 h = 7 h$
Write the final simplified expression:
$7 g + 4 h - 2 g + 3 h = 5 g + 7 h$

Common Mistake

Combining different letters

You cannot simplify $5 a + 3 b$ into $8 ab$ or $8 a$ . The terms have different variable parts, so they must stay separate.

Keeping the sign with the term

In algebra, the sign before a term belongs to that term.

For example, in $8 c - 5 d + 2 c - d$ , the terms are:

$8 c$
$- 5 d$
$2 c$
$- d$

That last term is negative $d$ , which means $- 1 d$ .

Example

Simplifying with negative terms

Simplify $10 a - 3 b + 4 a - 5 b$ .

Keep each sign attached to the term after it.
Collect the $a$ terms: $10 a + 4 a$ .
Collect the $b$ terms: $- 3 b - 5 b$ .
Simplify the $a$ terms:
$10 a + 4 a = 14 a$
Simplify the $b$ terms:
$- 3 b - 5 b = - 8 b$
Write the answer:
$10 a - 3 b + 4 a - 5 b = 14 a - 8 b$

Tip

Underline or circle groups

In the exam, you can lightly underline all the $a$ terms, then all the $b$ terms. Make sure the minus sign is included in the underline.

Multiplying algebra terms

When simplifying multiplication, multiply the numbers together and then multiply the letters.

In algebra, we usually do not write the multiplication sign between letters. So $\times b$ is written as $ab$ .

Example

Multiplying a number and a letter term

Simplify $\times 4x$ .

Multiply the numbers first: 6 times 4 is 24.
Keep the variable $x$ .
Write the simplified answer:
$\times 4x = 24x$

Example

Multiplying two algebra terms

Simplify $5a \times 2b$ .

Multiply the coefficients: 5 times 2 is 10.
Multiply the letters: $\times b = ab$ .
Put these together:
$5a \times 2b = 10ab$

Common Mistake

Adding instead of multiplying

$3a \times 4b$ is $12 ab$ , not $7 ab$ . When you see multiplication, multiply the coefficients.

Repeated letters and powers

Definition

Power

A power tells you how many times a number or variable is multiplied by itself. For example, $y^3$ means $\times y \times y$ .

The small raised number is called the index or exponent.

Common examples:

$\times b = b^2$
$\times t \times t = t^3$
$p^2 + p^2 = 2p^2$

Example

Writing repeated letters using powers

Simplify $\times r \times r$ .

Count how many times $r$ is multiplied by itself.
There are three $r$ factors.
Write this as a power:
$\times r \times r = r^3$

Example

Collecting squared terms

Simplify $4x^2 + 2x^2 - x^2$ .

Check that all terms have the same variable part, $x^2$ .
The coefficients are 4, 2 and -1.
Work out 4 + 2 - 1 = 5.
Put the $x^2$ back:
$4x^2 + 2x^2 - x^2 = 5x^2$

Common Mistake

Do not mix $x$ and $x^{2}$

$x$ and $x^2$ are unlike terms. For example, $3x + 2x^2$ cannot be simplified into $5x^2$ .

Dividing algebra expressions

Division can often be simplified by dividing the coefficients.

Example

Dividing a simple algebra term

Simplify $12h3\frac{12h}{3}$ .

The variable $h$ is being multiplied by 12.
Divide the coefficient by 3.
Since 12 divided by 3 is 4, the answer is:
$12h3=4h\frac{12h}{3} = 4h$

Sometimes you collect terms first, then divide.

Example

Collect first, then divide

Simplify $8y+4y2\frac{8y + 4y}{2}$ .

First collect the like terms in the numerator: $8 y + 4 y = 12 y$ .
Now divide by 2.
The simplified answer is:
$8y+4y2=6y\frac{8y + 4y}{2} = 6y$

Example

Cancelling the same letter

Simplify $15dd\frac{15d}{d}$ .

The $d$ on the top and the $d$ on the bottom cancel because they are the same factor.
This leaves 15.
So the simplified answer is:
$15dd=15\frac{15d}{d} = 15$

Putting everything together

Longer expressions usually need you to use more than one skill: collect like terms, keep signs carefully, and leave unlike terms separate.

Example

A mixed expression with letters and numbers

Simplify $6 a + 5 - 4 a + 3 b + 2 a - 8 + b$ .

Identify the different types of term: $a$ terms, $b$ terms and constants.
Collect the $a$ terms: $6 a - 4 a + 2 a$ .
Collect the $b$ terms: $3 b + b$ .
Collect the constants: 5 - 8.
Simplify each group:
$6a−4a+2a=4a3b+b=4b5−8=−3\begin{aligned} 6a - 4a + 2a &= 4a \\ 3b + b &= 4b \\ 5 - 8 &= -3 \end{aligned}$
Write the final expression:
$6 a + 5 - 4 a + 3 b + 2 a - 8 + b = 4 a + 4 b - 3$

Key Idea

Your final answer should be shorter

If your simplified answer is longer than the original expression, pause and check whether you have really simplified it.

Exam technique

In the exam

Read the whole expression before starting, especially if there are minus signs.
Collect only like terms: same letters and same powers.
For multiplication, multiply the numbers and write the letters next to each other.
For fractions, collect the numerator first if needed, then divide.
Check that your final answer has no unnecessary multiplication signs, such as writing $ab$ instead of $\times b$ .

Self review

Check yourself

Can you explain why $4 x + 3 y$ cannot be simplified to $7 x y$ ?
In $9 a - 2 b - 4 a + b$ , which terms are negative?
What is the difference between $x + x + x$ and $\times x \times x$ ?

Simplifying Algebra

What you'll learn

How to recognise like terms and collect them together.
How to simplify algebra using addition, subtraction, multiplication and division.
How to write repeated letters using powers, such as $x^2$ .
How to avoid common sign mistakes in one- and two-mark simplification questions.

What does “simplify” mean?

To simplify an algebra expression means to write it in a shorter, neater form without changing its value.

For example, if you see $x + x + x$ , you do not need to write the letter three times. You can write $3 x$ instead.

Definition

Expression, variable, term, coefficient

A variable is a letter that stands for a number, such as $x$ , $a$ or $n$ .
An expression is a calculation involving numbers, letters or both, such as $4 x + 7$ .
A term is one part of an expression, separated by plus or minus signs. In $5 x - 3 y + 2$ , the terms are $5 x$ , $- 3 y$ and 2.
A coefficient is the number multiplying a variable. In $6 m$ , the coefficient of $m$ is 6.

Key Idea

The main idea

Simplifying algebra is mostly about collecting matching terms and writing multiplication neatly.

Collecting the same letter

When you add the same letter term more than once, count how many of that term you have.

Think of $p$ as one $p$ . So:

$p + p$ becomes $2 p$ .
$p + p + p$ becomes $3 p$ .

Example

Collecting identical letter terms

Simplify $4 x + 5 x - 2 x$ .

Look at the letter in each term. Every term is an $x$ term, so they are like terms.
Collect the coefficients: 4, 5 and -2.
Work out 4 + 5 - 2 = 7.
Put the letter back on the answer:
$4 x + 5 x - 2 x = 7 x$

Common Mistake

Forgetting the invisible 1

A single $x$ means $1 x$ , not zero $x$ . So $x + x + x$ is $3 x$ , not just $x$ .

Terms with letters and numbers

Sometimes an expression has both letter terms and plain number terms.

A plain number term is called a constant because it does not change when the variable changes.

Definition

Constant

A constant is a term with no variable, such as 5, -3 or 12.

You can collect letter terms with the same letter, but constants stay separate from letter terms.

Example

A letter term and a constant

Simplify $k + k + 9$ .

The terms $k$ and $k$ are like terms.
Since $k + k = 2 k$ , replace those two terms with $2 k$ .
The 9 is a constant, so it cannot be combined with $2 k$ .
The simplified expression is:
$k + k + 9 = 2 k + 9$

Tip

Say it out loud

$2 k + 9$ means “2 lots of $k$ , plus 9”. Since you do not know what $k$ is, you cannot add the 9 onto the $2 k$ .

Like terms and unlike terms

Definition

Like terms

Like terms have exactly the same variable part. For example, $3 a$ and $7 a$ are like terms. But $3 a$ and $3 b$ are not like terms.

Here are some examples:

$5 x$ and $- 2 x$ are like terms.
$4 ab$ and $9 ab$ are like terms.
$x$ and $x^2$ are not like terms.
$6 m$ and $6 n$ are not like terms.

This picture shows the idea of sorting terms before collecting them.

Algebra terms sorted into like-term groups before simplifying

Example

Collecting two different types of term

Simplify $7 g + 4 h - 2 g + 3 h$ .

Group the $g$ terms together: $7 g - 2 g$ .
Group the $h$ terms together: $4 h + 3 h$ .
Simplify each group:
$7 g - 2 g = 5 g$
Simplify the other group:
$4 h + 3 h = 7 h$
Write the final simplified expression:
$7 g + 4 h - 2 g + 3 h = 5 g + 7 h$

Common Mistake

Combining different letters

You cannot simplify $5 a + 3 b$ into $8 ab$ or $8 a$ . The terms have different variable parts, so they must stay separate.

Keeping the sign with the term

In algebra, the sign before a term belongs to that term.

For example, in $8 c - 5 d + 2 c - d$ , the terms are:

$8 c$
$- 5 d$
$2 c$
$- d$

That last term is negative $d$ , which means $- 1 d$ .

Example

Simplifying with negative terms

Simplify $10 a - 3 b + 4 a - 5 b$ .

Keep each sign attached to the term after it.
Collect the $a$ terms: $10 a + 4 a$ .
Collect the $b$ terms: $- 3 b - 5 b$ .
Simplify the $a$ terms:
$10 a + 4 a = 14 a$
Simplify the $b$ terms:
$- 3 b - 5 b = - 8 b$
Write the answer:
$10 a - 3 b + 4 a - 5 b = 14 a - 8 b$

Tip

Underline or circle groups

In the exam, you can lightly underline all the $a$ terms, then all the $b$ terms. Make sure the minus sign is included in the underline.

Multiplying algebra terms

When simplifying multiplication, multiply the numbers together and then multiply the letters.

In algebra, we usually do not write the multiplication sign between letters. So $\times b$ is written as $ab$ .

Example

Multiplying a number and a letter term

Simplify $\times 4x$ .

Multiply the numbers first: 6 times 4 is 24.
Keep the variable $x$ .
Write the simplified answer:
$\times 4x = 24x$

Example

Multiplying two algebra terms

Simplify $5a \times 2b$ .

Multiply the coefficients: 5 times 2 is 10.
Multiply the letters: $\times b = ab$ .
Put these together:
$5a \times 2b = 10ab$

Common Mistake

Adding instead of multiplying

$3a \times 4b$ is $12 ab$ , not $7 ab$ . When you see multiplication, multiply the coefficients.

Repeated letters and powers

Definition

Power

A power tells you how many times a number or variable is multiplied by itself. For example, $y^3$ means $\times y \times y$ .

The small raised number is called the index or exponent.

Common examples:

$\times b = b^2$
$\times t \times t = t^3$
$p^2 + p^2 = 2p^2$

Example

Writing repeated letters using powers

Simplify $\times r \times r$ .

Count how many times $r$ is multiplied by itself.
There are three $r$ factors.
Write this as a power:
$\times r \times r = r^3$

Example

Collecting squared terms

Simplify $4x^2 + 2x^2 - x^2$ .

Check that all terms have the same variable part, $x^2$ .
The coefficients are 4, 2 and -1.
Work out 4 + 2 - 1 = 5.
Put the $x^2$ back:
$4x^2 + 2x^2 - x^2 = 5x^2$

Common Mistake

Do not mix $x$ and $x^{2}$

$x$ and $x^2$ are unlike terms. For example, $3x + 2x^2$ cannot be simplified into $5x^2$ .

Dividing algebra expressions

Division can often be simplified by dividing the coefficients.

Example

Dividing a simple algebra term

Simplify $12h3\frac{12h}{3}$ .

The variable $h$ is being multiplied by 12.
Divide the coefficient by 3.
Since 12 divided by 3 is 4, the answer is:
$12h3=4h\frac{12h}{3} = 4h$

Sometimes you collect terms first, then divide.

Example

Collect first, then divide

Simplify $8y+4y2\frac{8y + 4y}{2}$ .

First collect the like terms in the numerator: $8 y + 4 y = 12 y$ .
Now divide by 2.
The simplified answer is:
$8y+4y2=6y\frac{8y + 4y}{2} = 6y$

Example

Cancelling the same letter

Simplify $15dd\frac{15d}{d}$ .

The $d$ on the top and the $d$ on the bottom cancel because they are the same factor.
This leaves 15.
So the simplified answer is:
$15dd=15\frac{15d}{d} = 15$

Putting everything together

Longer expressions usually need you to use more than one skill: collect like terms, keep signs carefully, and leave unlike terms separate.

Example

A mixed expression with letters and numbers

Simplify $6 a + 5 - 4 a + 3 b + 2 a - 8 + b$ .

Identify the different types of term: $a$ terms, $b$ terms and constants.
Collect the $a$ terms: $6 a - 4 a + 2 a$ .
Collect the $b$ terms: $3 b + b$ .
Collect the constants: 5 - 8.
Simplify each group:
$6a−4a+2a=4a3b+b=4b5−8=−3\begin{aligned} 6a - 4a + 2a &= 4a \\ 3b + b &= 4b \\ 5 - 8 &= -3 \end{aligned}$
Write the final expression:
$6 a + 5 - 4 a + 3 b + 2 a - 8 + b = 4 a + 4 b - 3$

Key Idea

Your final answer should be shorter

If your simplified answer is longer than the original expression, pause and check whether you have really simplified it.

Exam technique

In the exam

Read the whole expression before starting, especially if there are minus signs.
Collect only like terms: same letters and same powers.
For multiplication, multiply the numbers and write the letters next to each other.
For fractions, collect the numerator first if needed, then divide.
Check that your final answer has no unnecessary multiplication signs, such as writing $ab$ instead of $\times b$ .

Self review

Check yourself

Can you explain why $4 x + 3 y$ cannot be simplified to $7 x y$ ?
In $9 a - 2 b - 4 a + b$ , which terms are negative?
What is the difference between $x + x + x$ and $\times x \times x$ ?

Simplifying Algebra

What you'll learn

What does “simplify” mean?

Collecting the same letter

Terms with letters and numbers

Like terms and unlike terms

Keeping the sign with the term

Multiplying algebra terms

Repeated letters and powers

Dividing algebra expressions

Putting everything together

Recap questions

You've reached the end

Simplifying Algebra

What you'll learn

What does “simplify” mean?

Collecting the same letter

Terms with letters and numbers

Like terms and unlike terms

Keeping the sign with the term

Multiplying algebra terms

Repeated letters and powers

Dividing algebra expressions

Putting everything together

Recap questions

You've reached the end

Simplifying Algebra

What you'll learn

What does “simplify” mean?

Collecting the same letter

Terms with letters and numbers

Like terms and unlike terms

Keeping the sign with the term

Multiplying algebra terms

Repeated letters and powers

Dividing algebra expressions

Putting everything together

Recap questions

You've reached the end

Simplifying Algebra Revision Guide

Simplifying Algebra

What you'll learn

What does “simplify” mean?

Collecting the same letter

Terms with letters and numbers

Like terms and unlike terms

Keeping the sign with the term

Multiplying algebra terms

Repeated letters and powers

Dividing algebra expressions

Putting everything together

Recap questions

You've reached the end

Simplifying Algebra Revision Guide